Skip to main content

All Questions

Filter by
Sorted by
Tagged with
4 votes
1 answer
96 views

Identifications between different phase spaces

I've discovered Adam's lecture notes on statistical mechanics after posting my first question about Minlo's discussion on continuous Gibbs measures. Adam's lecture notes are really good, but there is ...
MathMath's user avatar
  • 1,305
3 votes
0 answers
342 views

Sum of products of irreducible characters of the symmetric group over a subgroup

When trying to build a dual formulation for lattice gauge theories using Weingarten integration I am getting sums of the kind $$I^{m, n}_{\mu, \nu} (\sigma, \tau) = \sum_{\pi \in S_n} \chi_\mu (\pi \...
Volodymyr Chelnokov's user avatar
1 vote
1 answer
184 views

Measure, volume and cardinality on Minlos' book on statistical physics

The following content was based on Minlos' book on statistical physics. Let $\Lambda \subset \mathbb{R}^{d}$ be fixed (Minlos takes $d=3$ but I think the ideas follow without change to $d \ge 1$). We ...
MathMath's user avatar
  • 1,305
4 votes
2 answers
267 views

Grand-canonical Gibbs measure for continuous systems

Let's consider a bounded (maybe compact) set $\Lambda \subset \mathbb{R}^{d}$ with particles interacting on it. Suppose, for each $N \in \mathbb{N}$, $U_{N}: (\mathbb{R}^{d})^{N} \to \mathbb{R}\cup \{+...
JustWannaKnow's user avatar
0 votes
1 answer
86 views

Renormalization group map on hierarchical models

I have already addressed this problem on my previous question but I still have trouble understanding Brydges' RG maps on his lecture notes, so I'll try to elaborate my question a little better. Let $\...
JustWannaKnow's user avatar
2 votes
1 answer
161 views

Expected value of global functions in renormalization group

This is related to my previous question. I'm having some problems understanding the local to global program discussed in Brydge's lecture notes. We are assuming $C=C_{1}+\cdots+C_{N}$ is a covariance ...
JustWannaKnow's user avatar
2 votes
2 answers
294 views

Imprecise Definition of a $\sigma$-algebra

I'm reading some works on the hierarchical model in statistical mechanics and I came across an strange definition, which I need to clarify. Consider a finite set $\Lambda \subset \mathbb{Z}^{d}$. The ...
JustWannaKnow's user avatar
1 vote
1 answer
176 views

Gaussian Property of the Renormalization Group

Let $\Lambda \subset \mathbb{Z}^{d}$ be a finite set and $\varphi = (\varphi_{x})_{x\in \Lambda} \in \mathbb{R}^{|\Lambda|}$. Let $F^{\Lambda}=F^{\Lambda}(\varphi)$ be a real-valued global function, ...
JustWannaKnow's user avatar
2 votes
1 answer
1k views

Marchenko-Pastur Law under general covariance structure

Let $x_1,...,x_n\in\mathbb{R}^p$ be i.i.d. random vectors with mean 0 and covariance $\Sigma_p$. Let $S_{n,p}=\sum_{i=1}^nx_ix_i^T/n$ be the sample covariance. We consider the asymptotics of the ...
neverevernever's user avatar
0 votes
1 answer
301 views

Is there a Gaussian process for the solutions of the wave equation?

Call a Gaussian process $g$ a prior for a topological space $X$ if the realizations of $g$ are (a.s.) contained in $X$ and dense. Consider the 1D wave equation $\frac{\partial^2}{\partial t^2}u(t,x)=...
Markus Lange-Hegermann's user avatar
6 votes
0 answers
360 views

What is the status of the Born Rule in axiomatic QM?

While physicists have tried multiple times and failed to derive the Born Rule (for example: https://arxiv.org/pdf/quant-ph/0409144.pdf). I was wondering what axiomatic Quantum Mechanics had to say ...
More Anonymous's user avatar
7 votes
0 answers
579 views

Guises of the noncrossing partitions (NCPs)

From "Noncrossing partitions in surprising locations" by Jon McCammond: Certain mathematical structures make a habit of reoccuring in the most diverse list of settings. Some obvious ...
Tom Copeland's user avatar
  • 10.5k
3 votes
1 answer
2k views

Understanding Finite Size Scaling in Percolation Theory

Fundamental results in percolation theory are all based on the assumption that the system sizes are infinite, as the spanning/percolating cluster is by definition an infinitely sized cluster that ...
user929304's user avatar
4 votes
0 answers
164 views

List of Replica Symmetry results for different models?

Does anyone know of a good source that might have a list of problems or models along with what kind of replica symmetry they are conjectured to have? I am aware of some of the more famous results, e....
DJA's user avatar
  • 435
18 votes
0 answers
310 views

Profiles of very high dimensional functions

This question comes from trying to understand the recent success of deep neural nets. Neural networks just (crudely speaking) create a very complicated function of very many variables, and then ...
Igor Rivin's user avatar
  • 96.4k
6 votes
0 answers
334 views

Hints on an expository article about Kardar-Parisi-Zhang (KPZ)

It seems the KPZ is the next big thing in mathematical physics and probability. The skeletal idea is probably that while classical averages are in the Gaussian universality class, lots of other ...
Andrew Richards's user avatar
3 votes
0 answers
126 views

Other than Brownian motion, when else is it possible to define "normalized weighted infinite dimensional Lebesgue measure"?

In this article Sourav Chatterjee poses the question, how do we define the measure: $$d\mu(A)=\frac{1}{Z}\exp\left(-\frac{1}{4g^2}S_{YM}(A)\right)dA$$ The $Z$ here is an infinite normalizing ...
user avatar
2 votes
0 answers
103 views

Is this correct: Inflection points of Euler number graph in Island-Mainland transition correspond to spanning cluster site percolation threshold?

I'm writing with respect to the paper Khatun, Dutta, and Tarafdar - "Islands in Sea" and "Lakes in Mainland" phases and related transitions simulated on a square lattice. Here's a link to a PDF ...
user avatar
1 vote
0 answers
84 views

Particle density in phase space normalization under proliferation

Consider $1,..,N$ indistinguishable particles in $\mathbb{R}^2$ and let them evolve according to a brownian motion and proliferation. Let $u: \mathbb{R}_+ \times \mathbb{R}^2 \rightarrow \mathbb{R}_0^+...
Jack_Stiller10's user avatar
7 votes
3 answers
830 views

What is the link between the Domino Tilings and the Ising Model?

Is there a link between the theory of Domino Tilings and the Ising Model? In the global qualitative sense that physicists use, the answer is "yes". The connections could go like this: The dimer ...
john mangual's user avatar
  • 22.8k
21 votes
2 answers
981 views

What is the optimal speed to approach a red light?

Suppose from distance $d$, while driving at speed $v_0$, I notice that there's a red traffic light in front of me. Suppose that there are no other vehicles, my vehicle has perfect brakes, my maximum ...
domotorp's user avatar
  • 19k
5 votes
1 answer
945 views

Has a discrete/quantum theory of probability based on the Cournot-Borel principle or something been developed?

In 1930, Émile Borel, the father of measure theory together with his student Lebesgue and a world-class expert in probability theory, published a short note Sur les probabilités universellement ...
Fabrice Pautot's user avatar
4 votes
1 answer
234 views

Renyi's conditional probability fields and turbulence

I've come to the conclusion that what is universal, in the statistics of high Reynolds number turbulence of viscous incompressible fluids, could be modelled exactly only with Alfred Renyi's concept of ...
Jean Duchon's user avatar
  • 3,085
2 votes
1 answer
109 views

Interacting particle systems with spatially inhomogeneous hydrodynamic equations

Are there known examples of spatially inhimogeneous PDE appearing as hydrodynamic equations of interacting particle systems? In particular, I wonder whether a spatially inhomogeneous reaction ...
fourierwho's user avatar
7 votes
2 answers
626 views

What is the strongest known RSW result in planar percolation?

One of the weakest estimates conjectured to hold for critical planar percolation models (and proved in many cases) is the so-called RSW estimate. RSW estimate is the statement that the probability of ...
John Pardon's user avatar
  • 18.7k
35 votes
7 answers
6k views

Why is conformal invariance only possible for massless theories?

I'm conscious that this isn't necessarily a research level question, but I've asked this question on mathstackexchange, and received no answer. So I'm trying it here. A usual mantra in field theories ...
onamoonlessnight's user avatar
54 votes
4 answers
9k views

Why is Quantum Field Theory so topological?

I understand that my question suffers from my lack of knowledge about the field, but as a mathematician without much knowledge of physics I have been wondering much about the following and I always ...
A Physical newbie's user avatar
9 votes
1 answer
966 views

A necessary condition for differential entropy to be finite

An ensemble corresponding to a probability distribution usually has finite free energy so it is not a great loss of generality to assume that the ensemble also has finite energy in following ...
Henry.L's user avatar
  • 8,071
3 votes
1 answer
833 views

Sampling from a particular multivariate probability distribution

Given $3$ real variables $x_1, x_2, x_3 \equiv \bf{x}$, consider their probability density function (PDF) \begin{equation} P({\bf x}) = C \, p(x_1) \cdots p(x_3) \exp[f({\bf x})], \end{equation} where ...
James's user avatar
  • 343
4 votes
2 answers
2k views

Advanced reference and roadmap about random matrices theory

There is few posts on MO that asked about reference on this topic, and I found some difficulty during the process of getting myself into the subject so here is the question. I really want to hear ...
63 votes
3 answers
7k views

A roadmap to Hairer's theory for taming infinities

Background Martin Hairer gave recently some beautiful lectures in Israel on "taming infinities," namely on finding a mathematical theory that supports the highly successful computations from quantum ...
Gil Kalai's user avatar
  • 24.7k
11 votes
1 answer
626 views

Formula for $U(N)$ integration wanted

Before you jump on the "duplicate" buttom, let me say that I do not want to hear about Weingarten calculus and I do not want to see a character of the symmetric group. What I would like is a formula ...
Abdelmalek Abdesselam's user avatar
3 votes
0 answers
112 views

Uniqueness results for lattice spin systems (graphs)

Are there any nice uniqueness results for Gibbs-measures on lattice spin systems (graphs) that does not rely on Dobrushin's method?
Martinus Maximus's user avatar
1 vote
0 answers
94 views

Reason of the scaling factor $n^{2}$ in Hydrodynamic limits

In some books about hydrodynamic limits, example De Masi and Pressuti, when taking about the transition from micro to macro to get the hydrodynamic limit of some process it is mentioned that in order ...
Mario Antonio Ayala Valenzuela's user avatar
2 votes
0 answers
491 views

Is there a Bayesian theory of deterministic signal? Prequel and motivation for my previous question

This is a prequel to my question: What's the probability distribution of a deterministic signal or how to marginalize dynamical systems? (functional integrals in probability theory) Clearly my ...
Fabrice Pautot's user avatar
6 votes
2 answers
3k views

What's the probability distribution of a deterministic signal or how to marginalize dynamical systems? (functional integrals in probability theory)

Because I still have no idea how it is possible for me to write down seemingly important equations ... that don't make any sense (at least for me) and because I haven't got any helpful comment so far, ...
Fabrice Pautot's user avatar
7 votes
0 answers
497 views

Extreme unitary minimal models of conformal field theory

Some of the best understood conformal field theories are the 2D unitary minimal models $\mathcal{M}(m+1,m)$ indexed by the integer $m\ge 2$ and with central charge $$ c=1-\frac{6}{m(m+1)}\ . $$ I ...
Abdelmalek Abdesselam's user avatar
3 votes
0 answers
191 views

Infinite total variation of complex measure in Feynman path integral [closed]

I am trying to understand this: If one tries to define a Feynman path integral as a Wiener integral, then the complex measure could be of infinite total variation. What exactly does this mean? How ...
user87679's user avatar
4 votes
1 answer
229 views

How are the real-space RG transformations defined?

I'm reading Shang-keng Ma's book Modern theory of critical phenomena, and I'm a bit confused as to how the real-space RG transformations are defined. Ma basically says that these transformations are ...
Andrea Becker's user avatar
11 votes
2 answers
1k views

How should a mathematician approach the physics literature concerning percolation?

I would like to read some of the physics literature on two-dimensional percolation, however in attempting this I have run into two problems. (1) Physics papers on percolation are (relatively) hard ...
John Pardon's user avatar
  • 18.7k
4 votes
2 answers
271 views

Stationary distribution of last passage percolation

Consider last passage percolation model on $\mathbb{Z}^2$. I am interested to know if there is any known result for the stationary distribution of passage times, given some distribution for the ...
Austen's user avatar
  • 1,038
4 votes
0 answers
334 views

Unusual generalization of the law of large numbers

I have seen in physical literature an example of application of a very unusual form of the law of large numbers. I would like to understand how legitimate is the use of it, whether there are ...
asv's user avatar
  • 21.8k
5 votes
1 answer
365 views

power laws emerging from the sandpile model

Is there a rigorous proof that the abelian sandpile model generates a power law distribution of avalanche lengths?
Felix Goldberg's user avatar
6 votes
1 answer
353 views

Quaternion Wishart matrices of half-integer dimension?

For a physics application (quantum delay times of a chaotic scatterer) I need to generate $m$ positive random variables $\lambda_1,\lambda_2,\ldots\lambda_m$ with probability distribution $$P_\beta(\...
Carlo Beenakker's user avatar
0 votes
0 answers
42 views

Probability of close approach for multivariate normal variables

The following problem comes from a physical model of two groups of particles in three dimensions. I need to know the probability that the two groups of particles approach each other within some ...
John Jumper's user avatar
1 vote
1 answer
63 views

$P_{x}(T_{A}<\infty)<P_{x}(T_{B}<\infty)$ imply $Cap_{N}(A)<Cap_{N}(B)$, where $Cap_{N}$ is Newtonian capacity

We start a Brownian motion at $x\in [B(0,r)]^{c}$, where $B(0,r)$ is a large enough ball that contains compact sets $A$ and $B$. In other words, the B.M. starts on the exterior of $A$ and $B$. Then ...
Thomas Kojar's user avatar
  • 5,474
5 votes
1 answer
697 views

Harmonic Crystal using Random Walk

Me and my advisor were looking for a specific proof of the disorder in $2d$ harmonic crystals. We could not find a paper or a textbook with it, so I thought trying my luck here. Basically, it is a ...
Amir Sagiv's user avatar
  • 3,574
3 votes
1 answer
134 views

GOE convergence

As is well-known (at least in some circles), eigenvalue spacing distribution for large symmetric matrices converges as size goes to infinity (see this question for more background). The question is: ...
Igor Rivin's user avatar
  • 96.4k
5 votes
0 answers
139 views

Stationary point processes with arbitrarily slow decorrelation

A point process $P$ (a probability measure on simple, locally finite point configurations $\mathcal{C}$ on $\mathbb{R}$ - I'm restricting to the one-dimensional setting) is stationary when law-...
TLeble's user avatar
  • 121
1 vote
1 answer
143 views

Minimum of Random Energy Model (REM) with logarithmically correlated potential

In the paper [FB] (ArXiv, J. Phys. A), the authors analyse a particular Random Energy Model (REM) with logarithmically correlated potential and conjecture in Eq. (2) that the distribution function of ...
Eckhard's user avatar
  • 656